Assume that you have proved a complex-valued function $f$ to be equal to $e^x$ on the real line. Is it possible to conclude that $f$ is equal to $e^x$ on whole $\mathbb{C}$ and why / why not?
Maybe I do not see a simple solution. Since the real numbers do not lie dense in the complex plane, the only criterion I am able to remember cannot be applied.
In addition I am interested if the solution of my first question can be easily changed to fit for $e^{ax}$, $a \in \mathbb{R} \setminus \{0\}$. But I will think about it if I have an answer for the other question.
The question arose during an exercise lesson, noone including the trainer was able to answer it.
You can define $f(x)=0$ for all $x$ not on the real line to get a different extension. If there is no assumption on what type of extension you want there is no uniqueness.
However If $f$ is analytic on the complex plane and $f(x)=e^{x}$ for $x$ real then the same equation is true for all complex $x$ by the Identity Theorem.
See https://en.wikipedia.org/wiki/Identity_theorem